Narcissus was a greek heros from the area of Thespis in Boeotia. But the best-kown version of his tale was told by the roman poet Ovid, in his Metamorphosis.
Echo, a nymph, falls in love with a vain youth named Narcissus, who was the son of the blue Nymph Leirope of Thespia. The river god Cephisus had once encircled Leirope with the windings of his streams, and thus trapping her, had seduced the nymph. Concerned about the baby's welfare, Leirope went to consult the prophet Teiresias regarding her son's future. Teiresias told the nymph that Narcissus "would live to a ripe old age, as long as he never knew himself."
One day when Narcissus was out hunting stags, Echo stealthily followed the handsome youth through the woods, longing to address him but unable to speak first. When Narcissus finally heard footsteps and shouted "Who's there?", Echo answered "Who's there?" And so it went, until finally Echo showed herself and rushed to embrace the lovely youth. He pulled away from the nymph and vainly told her to leave him alone. Narcissus left Echo heartbroken and she spent the rest of her life in lonely glens, pining away for the love she never knew, until only her voice remained.
Eventually he became thirsty and went to drink from a stream. As he saw his reflection, he fell in love with it, not knowing that it was him. As he bent down to kiss it, it seemed to "run away" and he was heart broken. He grew thirstier but he wouldn't touch the water for fear of damaging his reflection, so he eventually died of thirst and self love, staring at his own reflection. The narcissus flower grew where he died.

Narcissus, Caravaggio, 1573-1610
Galleria Nazionale d'Arte Antica, Roma

Echo and Narcissus, John William Waterhouse, 1903
Walker Art Gallery, Liverpool

Avatar Arria Perreault looking at her face in a fountain in Second Life
Link to the story of Narcissus in the Metamorphosis of Ovidius (Book 3)